tutorial problems on ch.1 and 2[1]

750mm

1600mm

Ll45kN

Fig. P1.3 and P1.4

1200 N

.1 'ti'

il fc ,

I:~11

; IiI:Ii\,

(::

PROBLEMS

1.1 Two solid cylindrical rods AB and BC are welded together at Bandloaded as shown. Knowing that d, = 30 mm and d2 = 50 mm, find the aver-age normal stress in the midsection of (a) rod AB, (b) rod Be.

d1 125 kNA~+~ :==~·-n~~~~~~~~:60kN ~~~~~i.iiI!At-.\l~

I t 125~1-~ o.s m --- ----- 1.2 m ----~.I

Fig. P1.1 and P1.2

1.2 Two solid cylindrical rods AB and BC are welded together at B andloaded as shown. Knowing that the average normal stress must not exceed 150MPa in either rod, determine the smallest allowable values of the diameters d,and d2,

1.3 Two solid cylindrical rods AB and BC are welded together at Bandloaded as shown. Knowing that d, = 30 mm and d2 = 20 mrn find the normalstress at the midpoint of (a) rod AB, (b) rod Be.

1.4 Two solid cylindrical rods AB and BC are welded together at Bandloaded as shown. Knowing that the normal stress must not exceed 172 MFa ineither rod, determine the smallest allowable values of the diameters d, and d2.

1.5 A strain gage located at C on the surface of bone AB indicates thatthe average normal stress in the bone is 3.80 MPa when the bone is subjectedto two 1200-N forces as shown. Assuming the cross section of the bone at Cto be annular and knowing that its outer diameter is 2S mm, determine the in-ner diameter of the bone's cross section at e.

1.6 Two steel plates are to be held together by means of 6 mm-diame-ter high-strength steel bolts fitting snugly inside cylindrical brass spacers.Knowing that the average normal stress must not exceed 200 MPa in the boltsand 120 MPa in the spacers, determine the outer diameter of the spacers thatyields the most economical and safe design.

B "

!1200 N

Fig. P1.5 Fig. P1.6

18

Fig_ P1.7

1.7 Link BD consists of a single bar 30 mm wide and 12 mm thick.Knowing that each pin has a lO-mm diameter, determine the maximum valueof the average normal stress in link BD if (a) e = 0, (b) e = 90°.

1.8 Each of the four vertical links has an 8 X 36-mm uniform rectan-gular cross section and each of the four pins has a 16-mm diameter. Determinethe maximum value of the average normal stress in the links connecting (a)points Band D, (b) points C and E.

1.9 Two horizontal 22 kN forces are applied to pin B of the assemblyshown. Knowing that a pin of 20 mm diameter is used at each connection, de-termine the maximum value of the average normal stress (a) in link AB, (b) in

.link Be.

1.10 The frame shown consists offour wooden members, ABC, DEF,BE, and CF. Knowing that each member has a 50 X 100 mm rectangular crosssection and that each pin has a 12 mm diameter, determine the maximum valueof the average normal stress (a) in member BE, (b) in member CF.

.~A I~'---- 1.2 m---~. ~ 0.8 mi c

'LN1.0 m

LD f- 0.4 m -+<----

Fig. PUO

Problems 19

Fig. P1.8

12 rnrn

Fig. P1.9

20 Introduction-Concept of Stress

60mm

Fig. P1.13

Fig. P1.1S

~-

1.11 For the Pratt bridge truss and loading shown, determine the aver-age normal stress in member BE, knowing that the cross-sectional area of thatmember is 3780 rnrrr'.

350 kN 350 kN 350 kN

Fig. P1.11 and P1.12

1.12 Knowing that the average normal stress in member CE of the Prattbridge truss shown must not exceed 144 MPa for the given loading, deter-mine the cross-sectional area of that member which will yield the most eco-nomical and safe design. Assume that both ends of the member will be ade-quately reinforced.

1.13 A couple M of magnitude 1500 N . m is applied to the crank ofan engine. For the position shown, determine (a) the force P required to holdthe engine system in equilibrium, (b) the average normal stress in the con-necting rod BC, which has a 450-mm2 uniform cross section.

400mm

1,'>~I~:" \,1150 mm 200 mm

Fig. P1.14

1.14 Two hydraulic cylinders are used to control the position of the ro-botic arm ABC. Knowing that the controlrods attached at A and D each havea 20-mm diameter and happen to be parallel in the position shown, determinethe average normal stress in (a) member AE, (b) member DC.

1.15 The wooden members A and B are to be joined by plywood spliceplates which will be fully glued on the surfaces in contact. As part of the de-sign of the joint and knowing that the clearance between the ends of the mem-bers is to be 8 mrn, determine the smallest allowable length L if the averageshearing stress in the glue is not to exceed 800 kPa.

1.16 Determine the diameter of the largest circular hole that can bepunched into a sheet of polystyrene 6 mm thick, knowing that the force ex-erted by the punch is 45 kN and that a 55-MPa average shearing stress is re-quired to cause the material to fail.

P •• lol\~I-·-...p~

Fig. P1.17

1.17 Two wooden planks, each 22 mm thick and 150 mrn wide, arejoined by the glued mortise joint shown. Knowing that the joint will fail whenthe average shearing stress in the glue reaches 0.80 MPa, determine the small-est allowable length d of the cuts if the joint is to withstand an axial load ofmagnitude P = 5.4 kN.

1.18 A load P is applied to a steel rod supported as shown by an alu-minum plate into which a 15 mm-diameter hole has been drilled. Knowing thatthe shearing stress must not exceed 120 MPa in the steel rod and 70 MPa in thealuminum plate, determine the largest load P that may be applied to the rod.

1.19 The axial force in the column supporting the timber beam shownis P = 75 kN. Determine the smallest allowable length L of the bearing plateif the bearing stress in the timber is not to exceed 3.0 MPa.

Fig. P1.19

Problems 21

L140mm--1lOmm

T"15mm~;' •.

~6mmr

p

Fig. PU8

1.20 An axial load P is supported by a short W250 X 67 column ofcross-sectional area A = 8580 rnrn? and is distributed to a concrete founda-tion by a square plate as shown. Knowing that the average normal stress in thecolumn must not exceed ISO MPa and that the bearing stress on the concretefoundation must not exceed 12.5 MPa, determine the side a of the plate thatwill provide the most economical and safe design. Fig. P1.20


~" 12mm" t

Fig. P1.21

Fig, P1.22

1.21 Three wooden planks are fastened together by a series of bolts toform a column. The diameter of each bolt is 12 mm and the inner diameter ofeach washer is 16 mm, which is slightly larger than the diameter of the holesin the planks, Determine the smallest allowable outer diameter d of the wash-ers, knowing that the average normal stress in the bolts is 35 MPa and that thebearing stress between the washers and the planks must not exceed 8 MPa.

1.22 Link AB, of width b = 50 mm and thickness t = 6 mm, is used tosupport the end of a horizontal beam. Knowing that the average normal stressin the link is -138 MPa and that the average shearing stress in each of the twopins is 82 MPa, determine (a) the diameter d of the pins, (b) the average bear-ing stress in the link.

1.23 For the assembly and loading of Prob. 1.8, determine (a) the av-erage shearing stress in the pin at B, (b) the average bearing stress at B in linkBD, (c) the average bearing stress at B in member ABC, knowing that this mem-ber has a 10 X 50-mm uniform rectangular cross section.

1.24 For the assembly and loading of Prob. 1.8, determine (a) the av-erage shearing stress in the pin at C, (b) the average bearing stress at C inlink CE, (c) the average bearing stress at C in member ABC, knowing that thismember has a lOX 50-mm uniform rectangular cross section.

1.25 For the assembly and loading of Prob. 1.9, determine (a) the av-erage shearing stress in the pin at A, (b) the average bearing stress at A in mem-ber AB.

1.26 For the assembly and loading of Prob. 1.9, determine (a) the av-erage shearing stress in the pin at C, (b) the average bearing stress at C in mem-ber BC, (c) the average bearing stress at B in member Be.

1.27 Knowing that e = 40° and P = 9 kN, determine (a) the smallestallowable diameter of the pin at B if the average shearing stress in the pin isnot to exceed 120 MPa, (b) the corresponding average bearing stress in mem-ber AB at B, (c) the corresponding average bearing stress in each of the sup-port brackets at B.

p

16mm~~~


1.28 Determine the largest load P that may be applied at A whene = 60°, knowing that the average shearing stress in the 10-mm-diameter pinat B must not exceed 120 MPa and that the average bearing stress in memberAB and in the bracket at B must not exceed 90 MPa.

PROBLEMS

1.29 The 6-kN load P is supported by two wooden members of75 X 125-mm uniform rectangular cross section that are joined by the simpleglued scarf splice shown. Determine the normal and shearing stresses in theglued splice.

1.30 Two wooden members of 75 X 125-mm uniform rectangular crosssection are joined by the simple glued scarf splice shown. Knowing that themaximum allowable tensile stress in the glued splice is 500 kPa, determine (a)the largest load P which can be safely supported, (b) the corresponding shear-ing stress in the splice.

1.31 Two wooden members of 75 X 150-mm uniform rectangular crosssection are joined by the simple glued scarf splice shown. Knowing that themaximum allowable shearing stress in the glued splice is 0.62 MPa, determine(a) the largest load P which can be safely applied, (b) the corresponding ten-sile stress in the splice.

P'

~nm;;.=-[.~

p JSp'l:" 1,~-;;;i;::;~:;4,Jt~:'~~-r'.'.r--75mm '.


1.32 Two wooden members of 75 X 150 mm uniform rectangular crosssection are joined by the simple glued scarf splice shown. Knowing thatP = 10 kN determine the normal and shearing stresses in the glued splice.

1.33 A centric load P is applied to the granite block shown. Knowingthat the resulting maximum value of the shearing stress in the block is 17 MPa,determine (d) the magnitude of P, (b) the orientation of the surface on whichthe maximum shearing stress occurs, (c) the normal stress exerted on that sur-face, (d) the maximum value of the normal stress in the block.

P


1.34 A 1065 kN load P is applied to the granite block shown. Deter-mine the resulting maximum value of (a) the normal stress, (b) the shearingstress. Specify the orientation of the plane on which each of these maximumvalues occurs. Fig. P1.33 and P1.34

33


p


40° P

A ~~:::=;:::====;---;rT'~~'".0.38m

_1-~0.3m--1~0.45m


-,

I14m

28 kN

CFig. P1.41 and P1.42

. -!:."

1.35 A steel pipe of 300-mm outer diameter is fabricated from 6-mm-thick plate by welding along a helix which forms an angle of 250 with a planeperpendicular to the axis of the pipe. Knowing that a 250-kN axial force P isapplied to the pipe,' determine the normal and shearing stresses in directionsrespectively normal and tangential to the weld.

1.36 A steel pipe oflOO-mm outer diameter is fabricated from 6-mm-thick plate by weldingalong' a helix which forrris ari;angle of 250 with a planeperpendicular to the axis of the pipe. Knowing that the maximum allowablenormal and shearing stresses in directions respectively normal and tangentialto the weld are if::: 50 MPa and 'T ::: 30 MPa, determine the magnitude P ofthe largest axial-force that can be applied .to, the pipe.

1.37 Link BC is 6 nun thick, has a width w = 2S mm, and is made ofa steel with a 480-MPa ultimate strength in tension. What was the safety fac-tor used if the stlHcture shown was designed to support a 16-kN load P?

.. r-t30Qmm~

-...~t

.( p

D


1.38 Link BC is 6 mm thick and is made of a steel with a 450-MPa ul-timate strength in tension. What should be its width w if the structure shownis being designed to support a 20-kN load P with a factor of safety of 3?

1.39 Member ABC, which is supported by a pin and bracket at C and acable BD, was designed to support the 18 kN load P as shown. Knowing thatthe ultimate load for cable BD is 110 kN, determine the factor of safety withrespect to cable failure.

1.40 Knowing that the ultimate load for cable BD is 110 kN and that afactor of safety of 3.2 with respect to cable failure is required, determine themagnitude of the largest force P which can be safely applied as shown to mem-ber ABC.

1.41 Members AB and AC of the truss shown consist of bars of squarecross section made of the same alloy. It is known that a 20-mm-square bar ofthe same alloy was tested to failure and that an ultimate load of 120 kN wasrecorded. If bar AB has a lS-nun-square cross section, determine (a) the fac-tor of safety for bar AB, (b) the dimensions of the cross section of bar AC if itis to have the same factor of safety as bar AB.

1.42 Members AB and AC of the truss shown consist of bars of squarecross section made of the same alloy. It is known that a 20-mm-square bar ofthe same alloy was tested to failure and that an ultimate load of 120 kN wasrecorded. If a factor of safety of 3.2 is to be achieved for both bars, determinethe required dimensions of the cross section of (a) bar AB, (b) bar AC.

PROBLEMS

2.63 In a standard tensile test a steel rod of 22 mm diameter is subjectedto a tension force of 75 kN. Knowing that )J =: 0.3 and E =: 200 GPa determine(a) the elongation of the rod in an 200 mm gage length, (b) the change in di-ameter of the rod.

2.64 A standard tension test is used to determine the properties of anexperimental plastic. The test specimen is a IS-mm-diameter rod and it is sub-jected to a 3.5-kN tensile force. Knowing that an elongation of 11 mm and adecrease in diameter of 0.62 mm are observed in a I20-mm gage length, de-termine the modulus of elasticity, the modulus of rigidity, and Poisson's ratioof the material.

2.65 A 2-m length of an aluminum pipe of 240-mm outer diameter andlO-mm wall thickness is used as a short column and carries a centric axial loadof 640 kN. Knowing that E = 73 GPa and )J =: 0.33, determine (a) the changein length of the pipe, (b) the change in its outer diameter, (c) the change in itswall thickness.

fOkN(i:-~";~ .

Fig. P2.65

2.66 The change in diameter of a large steel bolt is carefully measuredas the nut is tightened. Knowing that E = 200 GPa and )J =: 0.33, determinethe internal force in the bolt, if the diameter is observed to decrease by 13 u.m,

2.67 An aluminum plate (E = 74 GPa, )J =: 0.33) plate is subjected to acentric axial load that causes a normal stress CT. Knowing that, before loading,a line of slope 2: J is scribed on the plate, determine the slope of the line whenCT == 125 MPa.

(22 mm diameter _

75 kN (.5kN• ! l~

1--200mm~

Fig. P2.63

p

ri20mm

Li5-111m diameter

Fig. P2.64

Fig. P2.66

Fig. P2.67

i'dji

I,I\.

.!

II

99

100 Stress and Strain-Axial Loading

Uy = 40MPa

A.~.L-l--'---L-.L,Br=-25 mm--

1

Fig. P2.69

y

Fig. P2.68

Ut = 80 MPa

... i"'- ••••••. 2670 N_.-":'::-:.:

A B

12mm

2.68 A 2670 N tensile load is applied to a test coupon made from1.5 mm flat steel plate (E = 200 GPa, v = 0.30). Determine the resultingchange (a) in the 50 mm gage length, (b) in the width of portion AB of thetest coupon, (c) in the thickness of portion AB, (d) in the cross-sectional areaof portion AB.

2.69 A 25 mm square is scribed on the side of a large steel pressure ves-sel. After pressurization the biaxial stress condition at the square is as shown.Knowing that E = 200 GPa and v = 0.30, determine the change in length of(a) side AB, (b) side Be, (c) diagonal AC.

2.70 For the square of Prob. 2.69, determine the percent change in theslope of diagonal DB due to the pressurization of the vessel.

2.71 A fabric used in air-inflated structures is subjected to a biaxial load-ing that results in normal stresses Ux = 120 MPa and a, = 160 MPa. Knowingthat the properties of the fabric can be approximated as E = 87 GPa andv = 0.34, determine the change in length of (a) side AB, (b) side Be, (c),diagonal AC.

I240 mm 600 mm

'JLBI~(IIIt:ill

D'

-,.1 I~50mm

Fig. P2.72

2.72 The brass rod AD is fitted with a jacket that is used to apply a hy-drostatic pressure of 48 MFa to the 2S0-mm portion Be of the rod. Knowingthat E = 105 GPa and v = 0.33, determine (a) the change in the total lengthAD, (b) the change in diameter of portion Be of the rod.

102 Stress and Strain-Axial Loading

180mm~


Fig. P2.80

2.77 Two blocks of rubber, each of width w = 60 mm, are bonded torigid supports and to the movable plate AB. Knowing that a force of magni-tude P = 19 kN causes a deflection 0 = 3 mm, determine the modulus of rigid-ity of the rubber used.

2.78 Two blocks of rubber, for which G = 7.5 MPa are bonded to rigidsupports and to the movable plate AB. Knowing that the width of each blockis w = 80 mm, determine the effective spring constant, k = Plo, of the sys-tem.

2.79 The plastic block shown is bonded to a fixed base and to a hori-zontal rigid plate to which a force P is applied. Knowing that for the plasticused G = 380 MPa, determine the deflection of the plate when P = 40 kN.

90mm1

Fig. P2.79

2.80 A vibration isolation unit consists of two blocks of hard rubberbonded to plate AB and to rigid supports as shown. For the type and grade ofrubber used 7all = 1.5 MPa and G = 12.5 MPa. Knowing that a centric verticalforce of magnitude P = 14 kN must cause a 2.5 mm vertical deflection of theplate AB, determine the smallest allowable dimensions a and b of the block.

2.81 An elastomeric bearing (G = 0.9 MPa) is used to support a bridgegirder as shown to provide flexibility during earthquakes. The beam must notdisplace more than 10 mm when a 22-kN lateral load is applied as shown.Knowing that the maximum allowable shearing stress is 420 kPa, determine(a) the smallest allowable dimension b, (b) the smallest required thickness Q.

Fig. P2.81

tutorial problems on ch.1 and 2[1]

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